On n-dependent groups and fields III. Multilinear forms and invariant connected components

Abstract

We develop some model theory of multilinear forms, generalizing Granger's work in the bilinear case. In particular, after proving a quantifier elimination result, we show that for an NIP field $K$, the theory of infinite-dimensional non-degenerate alternating $n$-linear spaces over $K$ is strictly $n$-dependent, and is NSOP${}_1$ if $K$ is. These results rely on a new Composition Lemma for functions of arbitrary arity and NIP relations (which in turn relies on certain higher-arity generalizations of the Sauer--Shelah lemma). We also study the invariant connected components $G^{\infty }$ in $n$-dependent groups, demonstrating their relative absoluteness.

arXiv commentv.3: 60 pages; the result on $G^{\infty }$ is generalized from abelian to arbitrary groups; many minor corrections and clarifications throughout the paper following referee's report